What is the Value of this Summation?

In summary, the value of summation is used to calculate the total sum of a series of numbers. This can be done manually or with a calculator/program. There is a formula for finding the value of summation, which involves the summation symbol and the upper and lower limits of the series. The value of summation can be negative depending on the numbers in the series. It has practical applications in various fields, such as finance, physics, and statistics.
  • #1
Albert1
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\[\sum_{n=1}^{9999}\frac{1}{(\sqrt{{n+1}}+\sqrt{n}\,\,)(\sqrt[4]{n+1}\,\,+\sqrt[4]{n}\,\,)}\]
 
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  • #2
Albert said:
$\sum_{1}^{9999}\dfrac{1}{(\sqrt{{n+1}}+\sqrt{n}\,\,)(\sqrt[4]{n+1}\,\,+\sqrt[4]{n}\,\,)}$

Rationalizing You have...

$\displaystyle \frac{1}{(\sqrt{n+1} + \sqrt{n})\ (\sqrt[4]{n+1} + \sqrt[4]{n})} = \frac{(\sqrt{n+1} - \sqrt{n})\ (\sqrt[4] {n+1} - \sqrt[4] {n})}{\sqrt{n+1} - \sqrt{n}} = \sqrt[4] {n+1} - \sqrt[4] {n}$

... that is a 'telescopic sum'...

$\displaystyle S= \sqrt[4] {2} - \sqrt[4] {1} + \sqrt[4] {3} - \sqrt[4] {2} + ... + \sqrt[4] {10000} - \sqrt[4] {9999} = 10 - 1 = 9$

Kind regards

$\chi$ $\sigma$
 
  • #3
well done !
 

FAQ: What is the Value of this Summation?

What is the purpose of finding the value of summation?

The purpose of finding the value of summation is to calculate the total sum of a given series of numbers. It is a common mathematical operation used in various fields such as statistics, physics, and engineering.

How do you find the value of summation?

The value of summation can be found by adding up all the numbers in the series. This can be done manually by writing out all the terms and adding them together, or it can be done using a calculator or computer program.

Is there a formula for finding the value of summation?

Yes, there is a formula for finding the value of summation. It is called the summation formula or the sigma notation. It involves using the summation symbol (∑) and the upper and lower limits of the series, along with the expression for the terms of the series.

Can the value of summation be negative?

Yes, the value of summation can be negative. It depends on the numbers in the series and their arrangement. If the series contains both positive and negative numbers, the total sum can be positive or negative. If the series only contains negative numbers, the total sum will be negative.

What are some practical applications of finding the value of summation?

The value of summation is used in various real-world applications, such as calculating the total cost of a project with multiple expenses, finding the average of a set of data, and determining the total force exerted on an object in physics. It is also commonly used in financial analysis and in determining probabilities in statistics.

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